Question

Dany jest wielomian W(x) określony wzorem W(x)= x4 + mx3 + (n-1)x2 + mx + 3, wiedząc że W(1)=W(-1) oraz W(2)= 27, wyznacz m i n .

Answer 1

W(x) = x^4 + m x^3 + ( n -1) *x^2 + m x + 3

W(1) = W( -1)   i  W(2) = 27

zatem

1^4 + m*1^3 + ( n -1)*1^2 + m*1 + 3 = ( -1)^4 + m*( -1)^3 + ( n -1)*( -1)^2 + m*(-1) + 3

1 + m + n - 1 + m + 3 = 1 - m + n - 1 - m + 3

2 m + n + 3 = - 2 m + n + 3

4 m = 0

m = 0

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W(2) = 2^4 + m*2^3 + ( n -1)*2^2 + m*2 + 3 =

= 16 + 8m + 4 n - 4 + 2 m + 3 = 10 m + 4 n + 15 = 27

Mamy

m = 0

10 m + 4 n = 12

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10*0 + 4 n = 12

4 n = 12

n = 3

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Odp.  m = 0 ,  n = 3

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